Atlas › Modular forms › Weight 2 › Level 388 › 388.2.a

Newspace 388.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(388))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
388.2.a.a	3	degree 3	$3q - 2q^{3} - 5q^{5} + q^{7} + O(q^{8})$	−
388.2.a.b	5	degree 5	$5q + 2q^{3} + 5q^{5} + q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
388.2.a.a	$x^{3} + 17x^{2} + 80x + 71$
388.2.a.b	$x^{5} - 17x^{4} - 18x^{3} + 1413x^{2} - 4444x - 8796$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(388))$	8
Old forms	39
Cusp forms $S_{2}(\Gamma_0(388))$	47
Eisenstein series $E_{2}(\Gamma_0(388))$	5
Modular forms $M_{2}(\Gamma_0(388))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(388)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(97))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(194))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.192 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 2, 4, 97, 194, 388]: [0, 1, 2, 8, 26, 52] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(388, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 388:2 1000

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